Algebra Examples

$x^{5} - x$ , $x^{5} - x^{2}$ , $x^{5} - x^{3}$

Step 1

Factor $x^{5} - x$ .

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Step 1.1

Factor $x$ out of $x^{5} - x$ .

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Step 1.1.1

Factor $x$ out of $x^{5}$ .

$x \cdot x^{4} - x$

Step 1.1.2

Factor $x$ out of $- x$ .

$x \cdot x^{4} + x \cdot - 1$

Step 1.1.3

Factor $x$ out of $x \cdot x^{4} + x \cdot - 1$ .

$x (x^{4} - 1)$

Step 1.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$x ({(x^{2})}^{2} - 1)$

Step 1.3

Rewrite $1$ as $1^{2}$ .

$x ({(x^{2})}^{2} - 1^{2})$

Step 1.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 1$ .

$x ((x^{2} + 1) (x^{2} - 1))$

Step 1.5

Factor.

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Step 1.5.1

Simplify.

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Step 1.5.1.1

Rewrite $1$ as $1^{2}$ .

$x ((x^{2} + 1) (x^{2} - 1^{2}))$

Step 1.5.1.2

Factor.

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Step 1.5.1.2.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$x ((x^{2} + 1) ((x + 1) (x - 1)))$

Step 1.5.1.2.2

Remove unnecessary parentheses.

$x ((x^{2} + 1) (x + 1) (x - 1))$

Step 1.5.2

Remove unnecessary parentheses.

$x (x^{2} + 1) (x + 1) (x - 1)$

Step 2

Factor $x^{5} - x^{2}$ .

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Step 2.1

Factor $x^{2}$ out of $x^{5} - x^{2}$ .

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Step 2.1.1

Factor $x^{2}$ out of $x^{5}$ .

$x^{2} x^{3} - x^{2}$

Step 2.1.2

Factor $x^{2}$ out of $- x^{2}$ .

$x^{2} x^{3} + x^{2} \cdot - 1$

Step 2.1.3

Factor $x^{2}$ out of $x^{2} x^{3} + x^{2} \cdot - 1$ .

$x^{2} (x^{3} - 1)$

Step 2.2

Rewrite $1$ as $1^{3}$ .

$x^{2} (x^{3} - 1^{3})$

Step 2.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 1$ .

$x^{2} ((x - 1) (x^{2} + x \cdot 1 + 1^{2}))$

Step 2.4

Factor.

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Step 2.4.1

Simplify.

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Step 2.4.1.1

Multiply $x$ by $1$ .

$x^{2} ((x - 1) (x^{2} + x + 1^{2}))$

Step 2.4.1.2

One to any power is one.

$x^{2} ((x - 1) (x^{2} + x + 1))$

Step 2.4.2

Remove unnecessary parentheses.

$x^{2} (x - 1) (x^{2} + x + 1)$

Step 3

Factor $x^{5} - x^{3}$ .

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Step 3.1

Factor $x^{3}$ out of $x^{5} - x^{3}$ .

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Step 3.1.1

Factor $x^{3}$ out of $x^{5}$ .

$x^{3} x^{2} - x^{3}$

Step 3.1.2

Factor $x^{3}$ out of $- x^{3}$ .

$x^{3} x^{2} + x^{3} \cdot - 1$

Step 3.1.3

Factor $x^{3}$ out of $x^{3} x^{2} + x^{3} \cdot - 1$ .

$x^{3} (x^{2} - 1)$

Step 3.2

Rewrite $1$ as $1^{2}$ .

$x^{3} (x^{2} - 1^{2})$

Step 3.3

Factor.

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Step 3.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$x^{3} ((x + 1) (x - 1))$

Step 3.3.2

Remove unnecessary parentheses.

$x^{3} (x + 1) (x - 1)$

Step 4

Since $x (x^{2} + 1) (x + 1) (x - 1), x^{2} (x - 1) (x^{2} + x + 1), x^{3} (x + 1) (x - 1)$ contains both numbers and variables, there are four steps to find the LCM. Find LCM for the numeric, variable, and compound variable parts. Then, multiply them all together.

Steps to find the LCM for $x (x^{2} + 1) (x + 1) (x - 1), x^{2} (x - 1) (x^{2} + x + 1), x^{3} (x + 1) (x - 1)$ are:

1. Find the LCM for the numeric part $1, 1, 1$ .

2. Find the LCM for the variable part $x^{1}, x^{2}, x^{3}$ .

3. Find the LCM for the compound variable part $x^{2} + 1, x + 1, x - 1, x - 1, x^{2} + x + 1, x + 1, x - 1$ .

4. Multiply each LCM together.

Step 5

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 6

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 7

The LCM of $1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 8

The factor for $x^{1}$ is $x$ itself.

$x^{1} = x$

$x$ occurs $1$ time.

Step 9

The factors for $x^{2}$ are $x \cdot x$ , which is $x$ multiplied by each other $2$ times.

$x^{2} = x \cdot x$

$x$ occurs $2$ times.

Step 10

The factors for $x^{3}$ are $x \cdot x \cdot x$ , which is $x$ multiplied by each other $3$ times.

$x^{3} = x \cdot x \cdot x$

$x$ occurs $3$ times.

Step 11

The LCM of $x^{1}, x^{2}, x^{3}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x \cdot x \cdot x$

Step 12

Simplify $x \cdot x \cdot x$ .

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Step 12.1

Multiply $x$ by $x$ .

$x^{2} \cdot x$

Step 12.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 12.2.1

Multiply $x^{2}$ by $x$ .

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Step 12.2.1.1

Raise $x$ to the power of $1$ .

$x^{2} \cdot x^{1}$

Step 12.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2 + 1}$

Step 12.2.2

Add $2$ and $1$ .

$x^{3}$

Step 13

The factor for $x^{2} + 1$ is $x^{2} + 1$ itself.

$(x^{2} + 1) = x^{2} + 1$

$(x^{2} + 1)$ occurs $1$ time.

Step 14

The factor for $x + 1$ is $x + 1$ itself.

$(x + 1) = x + 1$

$(x + 1)$ occurs $1$ time.

Step 15

The factor for $x - 1$ is $x - 1$ itself.

$(x - 1) = x - 1$

$(x - 1)$ occurs $1$ time.

Step 16

The factor for $x^{2} + x + 1$ is $x^{2} + x + 1$ itself.

$(x^{2} + x + 1) = x^{2} + x + 1$

$(x^{2} + x + 1)$ occurs $1$ time.

Step 17

The factor for $x + 1$ is $x + 1$ itself.

$(x + 1) = x + 1$

$(x + 1)$ occurs $1$ time.

Step 18

The factor for $x - 1$ is $x - 1$ itself.

$(x - 1) = x - 1$

$(x - 1)$ occurs $1$ time.

Step 19

The LCM of $x^{2} + 1, x + 1, x - 1, x - 1, x^{2} + x + 1, x + 1, x - 1$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(x^{2} + 1) (x + 1) (x - 1) (x^{2} + x + 1)$

Step 20

The Least Common Multiple $LCM$ of some numbers is the smallest number that the numbers are factors of.

$x^{3} (x^{2} + 1) (x + 1) (x - 1) (x^{2} + x + 1)$